The centre of our galaxy: a holostar

L/upd: 2019/05/23The article argues about Sagittarius A*, the supermassive black hole in our Milky Way, and is inspired by the revolutionary pic due to the Event Horizon Telescope activity. We define the outlines of our theory, which describes Sgr A* as a holostar, as well as we give a proper model in order to frame the inertial equivalence in space-rotating systems.

As anticipated in my previous article, we succeeded in getting the direct observation, for the first time, of what we can define the “shadow” of the event horizon of a black hole, to be exact of the gigantic and supermassive black hole M87*, which is located in the center of the namesake galaxy M87 (or Virgo A) . The relative pic was shared around the world and was rightly named “the pic of the century”. The team of scientists of the EHT (Event Horizon Telescope) project, announced this epochal novelty on April 10, 2019, together with the release of the pic and the publication of a series of highly detailed scientific articles. Let’s recap: the VLBI technology is the networking of the largest radio telescopes, located in various parts of the globe, so as to form the equivalent of a single radio telescope (Event Horizon Telescope or EHT) with a radius equal to that of the Earth. This technology has lead us to the result, the pic of the century. We will better discuss the results released by EHT team to the world on April 10th 19 in the next article.

However EHT had already provided a public result, relative to what seemed to be the main object of its efforts, i.e. Sagittarius A*, the supermassive black hole in our galaxy. Honestly, everyone expected the event of April 10th to deal with it even more, given that the discussion started with that early result is still pending. The difference, which in all likelihood has orientated the choice of the EHT team, with M87* is that Sagittarius A* hasn’t shown any signs of an accretion disk during this period. An accretion disk is made up of matter that, whirling and at extreme speed, falls into a black hole: while doing this matter produces light radiation in the form, for us remote observers, of radio waves. It is precisely this light, which rings it, that allows us to see the shadow of a black hole, by contrast, as a background. Now, if we start from the assumption that every black hole is surrounded by an event horizon that prevents even light escaping, only this boundary light, the last sign of life of matter before disappearing beyond the horizon itself, can give us an image that, like a shadow or a background, allows us to glimpse the black hole. Without an accretion disk, any black hole, being surrounded by that insuperable horizon, remains totally invisible. In this regard I used the comparison of the Sun and its corona (an area which surrounds our star and is much warmer than the solar surface). If I take an x-ray picture, the Sun appears black and the corona bright; so, if having only x-rays available, I could see the surface of the star only by contrast with its corona. Therefore, using only x-ray detectors while I were looking for stars to photograph, I would prefer those stars with a corona, like the Sun. But what happens if I’d have not only x-rays, but also visible light available? If I use the latter, it happens that I see the Sun, i.e. its surface, without being dazzled by the corona. Something like this has happened, and I suppose it’s still happening on EHT’s computers, about Sagittarius A*. With very unexpected results. The EHT team, as I said, before the recent silence, had already provided a result (with relative pic of Sagittarius A*). More exactly, so far it has observed that “Its (Sgr A*’s) emission region is so small that the source may actually have to point directly at the direction of the Earth” and “it could be that the radio jet is pointing almost at us” so “we are looking at this beast from a very special vantage point”. Here the links: https://www.nanowerk.com/news2/space/newsid=51934.phphttps://arxiv.org/abs/1901.06226

So, as you can see from the wonderful images shown in the articles, there is a substantial difference between what we expected to see (jets of matter, accelerated almost to the speed of light, would have allowed us to see, as their background, the gigantic black silhouette, the shadow, of the horizon of Sgr A*) and what, at least at the moment (i.e. without a true accretion disk), we have seen: an uniform emission zone. Hence: either there is matter that is falling on Sgr A* in a symmetrical and uniform way, like dew, something quite incredible, or what we are seeing is rather a jet. Jets of matter may occur as a result of a funnel effect: in other words, when there is an excess of matter that is falling, swirling too much into a too small area, a funnel effect is produced, until a part is expelled. Now a jet, to be seen so small and symmetrical, has to point directly to the Earth, with a really remarkable precision. Obviously, such a symmetrical emission should have a polar origin, which would imply that the supermassive black hole is spinning “lying” with respect to the galactic plane of rotation (as Uranus does with respect to the plane of rotation of planets in the solar system). The fact that one of its poles could point with extreme precision towards us has the same probability of occurring than the one of winning in the national lottery with the first ticket which we decide to bet on. It would appear as if Galileo, after having ingeniously created an adequate telescope, had pointed it at the Moon for the first time and, first of any other thing, had observed that on the lunar soil there was a telescope pointed exactly towards him. Not to mention that a jet of matter “shot” towards us at a speed close to that of light should show a considerable blueshift effect.

So, the conclusion I consider more probable, the one that, in my opinion, will be corroborated by the results of new observations, once they will be provided, is that we are looking simply at what it seems to be, i.e. at an image of that winking superstar, not at an event horizon (see Footnote 1). (Image still enlarged, but I suppose that this could depend on the radio wavelength chosen for each single round of observations: if the next one chosen will be smaller – or the frequency higher – I think we will see a smaller zoom effect).

As I wrote in my main article, the EGR (the extension of General Relativity from uniformly accelerated systems to space-rotating systems) predicts the theoretical impossibility of any gravitational singularity, and therefore the non-existence of any event horizon. The reason is largely intuitive, since EGR is a theory that implies a new (and more relativistic) conception of inertia: just as the inertia of a massive body prevents it from reaching a speed equal to that of light, in the same way its inertia prevents it from reaching a zone of space wherein escape velocity is greater than or equal to that of light, because, to get there, the body would have to reach a speed greater than or equal to that of light. Whatever the condition, free fall, as in the example of the first article, or high density as in the common case of collapsing of a neutron star, the single particle P, approaching the most extreme state of curvature, observes a corresponding increase of the remote inertia, in particular of the relativistic mass, of all other particles, together with their relative temporal slowing down: therefore their motion looks as lagging behind the one of particle P, and as balancing its growing proper inertia in delay. The outward push, which is experienced by the P particle, due to extreme relativistic conditions of other ones, gets at some point to an equilibrium with the gravitational pressure on P. Relativistic inertial mass is relative, in contrast to gravitational one ,which is not. Therefore the relative times, for this inertial component (which is increasingly significant in the neighbourhood of the Schwarzschild radius – see Fn 2) have to be treated as in Restricted Relativity, since relativistic inertial mass does nothing but progressively replace the velocity component, when acceleration acts in conditions close to the speed of light. In particular, relativistic inertial mass is always remote for any single particle, as concerning all the others, and it is not increasing the particle gravitational mass, but its proper inertia, or reverse acceleration, which is a centrifugal component. All this has the consequence that the motion of all the particles finally lies on the spheroidal surface about which we argued in the first article, whose smallest or polar radius is equal to the Schwarzschild radius.

From the point of view of the fast-ticking observer (see Fn 3), he/she sees the situation as that relative to the rotating astronaut hanging on to a handle (see as before “Einstein is asked etc.”). That is, he/she sees a constant centrifugal inertial component undergone by the particle P (increasing proper inertia: if we could use the comparison of a proton in an accelerator, he/she sees the proton acquiring more and more inertial mass and therefore pressuring towards the outer edge of the accelerator, tending to describe a longer circumference as a result of its greater inertia). From the point of view of the single particle P, it sees the situation as if it were the remote inertia, that of all the other particles, to grow; but, due to consequent relativistic dilation of their time, their pressure acts ever more delayed, letting an increasing part of P’s acceleration balance gravitational pressure (using the previous comparison, our proton, from its own point of view, sees the accelerator acquiring greater relativistic mass and consequently time dilation, that is, it sees the latter describing a circumference with radius ever greater, which leaves our proton free to put more pressure on the edge, on equal proper inertia).

From the most “remote” point of view, i.e. the ideal center of rotation, the situation is shown better in this way: it’s as if a retrograde precession (with respect to astronaut’s rotation) of space is implying for all the particles, as a consequence, to be describing circumferences that lag behind, or end geometrically first, on equal radius, with respect to the euclidean ones (the ones we all know, related to a flat space). Gravitational curvature causes the opposite effect, to make particles describe circumferences that move forward, or end geometrically later, on equal radius (well effect.. imagine you are going down a spiral staircase: to find yourself on equal than starting coordinates, for example on the y-axis, you tread a path longer than a flat circumference, on equal radius, but if the y-axis is rotated in the opposite direction to yours, you end the path by treading a distance shorter than a flat circumference, on equal radius). We can represent the same with better words by saying that, seen from the remote point of view, or center of rotation, gravitational curvature tends to make particles move following a tangent-to-the-radius angle that is smaller than the right one, while relativistic inertial mass due to relativistic speed tends to make particles move following a tangent-to-the-radius angle that is greater than the right one. Equilibrium is reached just along right angle, at a speed really close to that of light; however no particle ever exceeds it. All this is nothing other than the result of discovering, through the hypothesis of space rotation, the role of remote inertia, and the relative nature of its relationship and opposition to proper inertia.

Since we talked about spiral staircases and tangent angles, it is appropriate to introduce now the logarithmic spiral model (or “spira mirabilis” model) as the descriptive model that is proper for space-rotating systems.

After defining a generic logarithmic spiral as the locus of the points traced by a body P in a rectilinear and uniformly accelerated motion on a growing segment OP, while the segment OP is rotating (at a constant angular velocity ω) around the origin O of a system of polar coordinates (therefore the fast-ticking point is at a distant and polar position), we have that the distance r of P from the origin is developed in this way (here and below, “e” is Euler’s number, while “^” indicates the beginning of the exponent):r=ae^bθ (Note that from the point of view in O, or origin or remote point, P always moves in a rectilinear and uniformly accelerated motion – or free fall – whilst it is the x,y coordinate system that is rotating in the opposite direction; instead the fast-ticking point of view, as already shown in the first article, sees the movement of the astronaut as having completely absorbed the movement of the wheel). Among the many notable features of a logarithmic spiral, it is to mention that its tangent is constant at every point. By adapting the generic function to the motion of galaxies, we have:r= S e^pαθwhere S is the Schwarzschild radius of the galaxy, p is a dimensionless constant, α=dτ/dT=1⁄γ is the inverse of the Lorentz factor (see Fn 4). (Interesting, in my opinion, and perhaps in the one of some passionate about mathematics and symmetry mind, to note that p≅1⁄e if we take the crude average of the results of these two studies:https://arxiv.org/abs/astro-ph/0605728https://arxiv.org/abs/astro-ph/0005241average giving us a “pitch” angle (the complement of the tangent angle) equal to 20.38° – it’s to remark that for a pitch close to 20.20° we have exactly p=1⁄e). In order to complete: θ is the cumulative measurement of the angle between the rotating OP segment and the abscissa axis, so θ=ω∆T , with ω=f(Tc) function of the cosmological time or “age” of the Universe Tc (dimensionless quantity, in our context).

The function r= S e^pαθ so describes the isoequivalence curve for any massive body in a flat gravitational field, or, better, in a field characterized by an absolutely constant (spread in a constant way) mass density, and wherein dτ/dT=constant≅1 ; that’s to say our curve being the locus of the points where proper inertia and remote inertia are opposite and equal, so that the curve can be trod indifferently in both directions, in free fall. This is the real meaning of the aforementioned emergency of the principle of equivalence; as well as the misunderstanding of its emergency (i.e. about real nature of inertia) causes the apparent need for dark matter about which everyone is talking till now. The difference with the actual motion of a star is obviously due to gravity, namely local and overall density differences, as well as the collapse of density at the edge of the disk, with the passage to galactic halo zone. It is gravity also to make not indifferent the two directions of free fall, as well as to make all stars tread less “pitched” curves, so causing density waves and well-known spiral arms.

The relatively little density differences in close-to-collapse line of stellar matter, suggest us that a similar logarithmic spiral can describe the pre-collapsing particles movement in a star. In fact a collapsing neutron star is just coming from a state characterized by ERDM, extremely relativistic degenerate matter, in which the speed of particles is already close to the speed of light; so, if the star is isolated, while it is going to collapse, the equation of the combined spiral covered by deconfined particles (basically quarks), still in polar coordinates, is seen, from the fast-ticking point of view (while the astronaut, i.e. the candidate black hole, has absorbed upon him/herself, in the opposite sense, all the space rotation between him/herself and the f.t. point) to be:r= s e^(δ/D)pαθ with s Schwarzschild radius of the neutron star (which also describes the polar radius of the maximum pressure spheroidal surface inside the star – i.e. what we called center of mass so far); δ is the maximum density allowed by the Pauli exclusion principle, and D the actual stellar density in close-to-collapse line; increasing gravitational pressure is involving that δ/D is going to be close to 1, so that deconfined particles find ever less dispersive, in terms of energy, follow in a combined way the spiral; θ (see also below) is a cumulative quantity in the opposite sense outside or inside the s radius, so that collapsing toward s is, from a certain point on, the less expensive solution for particles; p is small as a consequence of density differences at different radii of the star (gravitational curvature). When collapse starts, for r→s in both senses (from outside and from inside) we have that the ratio between proper time and f.t. observer’s time approaches zero, α=dτ/dT→0so that the combined spiral trod by the particles under pressure degenerates into a circumference of radius s (polar projection of the real spheroidal surface) in which matter remains captured indefinitely.

We can describe the situation according to the famous example of running dogs, example which has always been reported as one of the intuitive way to understand logarithmic spirals: a few dogs, starting at the same time from various points on the edge of a huge circular room, and running because each of them has as objective getting to reach the one on its right, are describing, in a fairly intuitive way, each a logarithmic spiral. But if when approaching the center of the large room, each of them sees the dog on its own right becoming progressively bigger, heavier and slower, everyone will tend, in order to reach the other, more and more to maintain a right angle with respect to the radius that connects it to the center, describing, to the end, just a circumference.

I add, as regards neutron stars, that the characteristic, according to which points with maximum density, and therefore where the star is also maximally neutral and compact, lie on a spheroid at some km from the geometric center of the star, is a hypothesis by itself sufficient to explain very strong magnetic fields, otherwise incomprehensible. In fact, not only components different from neutrons would concentrate both on the surface and in the inside, but they, own to the very high spin and compactness of the star, would be trapped and forced to rotate at different radial speeds, with constant and huge consequent flows, generated in the hottest central zone.

In the case regarding our friend Sgr A* (I like to think to this superstar as a female entity, for the decisive matriarchal role played towards the entire galaxy) the situation is different, in the sense that it stands beyond S, the galactic Schwarzschild radius. It can be shown that the logarithmic spiral we have formulated (r= S e^pαθ) proceeds, also in this case, in its development with close continuity; however its tangent begins to change very slowly (but constantly), and its pitch begins to become gradually smaller, as the spiral winds, gaining proximity to the superstar, that is, the closer that dτ/dT becomes smaller than 1 and nearer to zero. Moreover, we note that θ becomes θ=-ω∆T, i.e. geometrically the segment OP (from origin to body P) appears to rotate in an inverse sense (time-verse rotation vs previous space-verse one); so, geometrically, θ becomes a cumulative quantity in the opposite sense. All this indicates that, from S on, matter can thicken in a single sense, and only following the path that leads to the superstar, as we wrote in the main article. The fact that there is only one central area of density and the related and progressive temporal distortion put an end to the relative isoequivalence of the two directions of the path, preserving it only for a single sense, the one of the arrow of time, i.e. the one which proceeds towards the center; in parallel, it disappears every motion difference due to gravity, i.e. relative to the mass the body P in free fall.

In other respects, as in stellar case, for r→s (s Schwarzschild radius of the superstar) the ratio between proper time and f.t. observer’s time approaches zero,α=dτ/dT→0so that the spiral degenerates, in the same way, in a circumference of radius s where matter remains captured – if we consider its development on the entire spheroidal surface – as in an almost perfectly two-dimensional and gigantic hologram, which emits energy almost uniformly, in the form of photons with large gravitational redshift (radio waves). The density of this holostar, in the case of Sgr A*, is consistent with the existence of hadrons and free electrons.

It is interesting to see the spiral inside the galactic center of mass, while developing by gathering its radial component in the opposite and time-verse direction, to be, except for the final stretch (heavily influenced by the relativistic dilation of time), the exact scale reproduction of the outer spiral (see Fn 5). This is one of the properties of the “spira mirabilis“, but in the case of our galaxy it gives rise to a particular harmony or resonance. We know that the galactic Schwarzschild radius is about 0.25 light years, 2.4 × 10 ^ 15 meters. That of Sgr A*, like its mass, is a part in 200,000, that is 1.2 × 10 ^ 10 meters. If we keep the proportion 1.2 × 10 ^ 10 : 2.4 × 10 ^ 15 = 2.4 × 10 ^ 15 : x, we have x = 4.8 × 10 ^ 20 meters, equal to 50.700 light years, which strangely is the radius of the galaxy. In fact, multiplied by two it gives us the diameter we all know, about 100,000 light years. Really “mirabilis“! See: https://en.wikipedia.org/wiki/Schwarzschild_radius#Parameterswith regard to the two Schwarzschild radii, while the scale derivation of the radius of the entire galactic disk, as far as I know, is mine. This shows, at least for the Milky Way, that r= S e^pαθ and r= s e^pαθ are the same identical curve, as in geometric theory of logarithmic spirals (so called self-similarity, which is so important in fractal geometry), and that switching the two Schwarzschild radii is an uniform scaling transformation of the same object, as changing the zoom on Google Maps. I believe that all this could be due to the fortunate position of the Milky Way, located in a very very peripheral cluster with respect to the reference super cluster (Virgo), and furthermore in a cluster, ever the local one, with a “handlebar” distribution of the mass, where the Milky Way represents the center of one of the two polarities. I think it is the best position to detect such a perfect “spira mirabilis“.

The center of the other pole of the local cluster handlebar, i.e. the Andromeda galaxy, was recently (meant on a scale of millions of years, of course) disturbed by the transit of the Triangulum galaxy (which, obviously, being smaller, suffered much more). However: the central superstar of Andromeda seems to have a mass 40 times that of ours, whilst the radius of the entire galaxy is 2.2 times that of ours. If the previous proportion has a certain value also for Andromeda, therefore the Andromeda mass should be √88 times that of ours, i.e. about 9.4 times. Having read estimations ranging from 1 time to 20 times, I would say that the result lies in a fairly middle position.

I apologize for the absolutely bad quality of equation rendering, unfortunately if I want to change it I will have to upgrade my subscription to the site, and then to insert necessary plugins. It was not possible writing subscripts, whilst I remind you that “^” indicates that what you read next is the exponent.

I thank Maurizio Bernardi for his contribution about logarithmic spiral geometry.

Footnotes:1) Scientist of EHT team, in their first article dated 19/04/10, state that their result is absolutely consistent with the Kerr relativistic metric (the GR metric relative to a spinning black hole), but that there are at least two other solutions, beyond that which foresees an event horizon; these comply with expectations of the Kerr metric, however without foreseeing a horizon; solutions which the result of April 10th cannot therefore exclude. As we will see in my next article, my suggestion, describing black holes as holostars, is the fourth hypothesis, and it is the one that requires coherence with the Kerr relativistic metric par excellence.2) The Schwarzschild radius is the theoretical minimum radius of the sphere within which all the mass of a body should be compressed in order to have an escape velocity on body’s surface equal to or greater than the speed of light; if the body proceeds in its collapse, the Schwarzschild radius remains the radius of the sphere that delimits the event horizon, since no information can come out from within that sphere.3) The fast-ticking observer is, in General Relativity, the inertial observer, the one not subject to any acceleration, located very far from the gravitational field, and, in the case of the EGR, placed in polar position, that is, not rotating; his/her clock, as noticing the relative slowness of all other clocks, is fast-ticking, i.e. with the faster ticking; consequently, his/hers is the clock that records the longest duration relatively to an event.4) The Lorentz factorγ is here defined simply as the ratio between the time marked by the fast-ticking clock of a non-accelerated observer in measuring the duration of an event, and proper time, that is the time marked by a clock joined with the event whose duration is measured: dT/dτ=γ. The factor is derived from the simple observation that the two clocks must measure the same speed, c, in the case of light. Here we assume the notation α used for its inverse, so α=1⁄γ=dτ/dT, that is α is the ratio between the proper time and the non-accelerated observer time.5) Actually, inner spiral, including final stretch, is the perfect scale reproduction of outer one, but in a four-dimensional scale, since we have to take into account space-time curvature. The path followed by P is exactly in scale, but it is trod along a spiral staircase, thus, in order to keep the scale, ends before if compared to a flat spiral: it is trod up to s instead than up to the geometric center, as the difference is trod through time.

Published by giorgio mistenda

CT th./ph. researcher trying to extend General Relativity (GR) from uniformly/accelerated systems to space/spinning ones. Hoping to do some cheerful science, too. Living in Sicily.
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